Semiclassical resonances for a two-level Schr\"odinger operator with a conical intersection

TL;DR

This paper investigates the resonant states of a two-level Schrödinger operator with a conical intersection, deriving a quantization condition and describing the resonance set asymptotically as a distorted lattice.

Contribution

It introduces a method to analyze resonances in a two-level Schrödinger operator with conical intersection using exact WKB solutions and provides a generalized quantization condition.

Findings

Derived a generalized Bohr-Sommerfeld quantization condition.

Provided an asymptotic description of the resonance set as a distorted lattice.

Connected local WKB solutions to global resonant states.

Abstract

We study the resonant set of a two-level Schr\"odinger operator with a linear conical intersection. This model operator can be decomposed into a direct sum of first order systems on the real half-line. For these ordinary differential systems we locally construct exact WKB solutions, which are connected to global solutions, amongst which are resonant states. The main results are a generalized Bohr-Sommerfeld quantization condition and an asymptotic description of the set of resonances as a distorted lattice.